Eevee: Light sources appear as bright orbs in reflections
I solved the issue and am happy with the results, even if this is technically working around it. The core problem was that I was using a huge light radius: You want the sphere of the point light to encompass and represent only the bulb. I set it to 0.01 instead of the default 0.25: This not only results in the correct shadow softness, but the white orb is now just a tiny speck, precisely how a bright bulb should appear in a reflection! It's also hidden behind a lamp cover so it shines even this way only when seen from the right angle.